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) 1 , (    / l oebr2 2007 2, November n non Gournay Antoine ∞ p , ,btlo at look but ), = ) m k n 0 ε B ε ( ( 1 l X ∈ p p ( { if if if , , n 0 R ε d ) )) , = ) > etestgvnb h ntbl in ball unit the by given set the be 1 2 1 0 ,..., n rmaoeb min by above from and ( hnw a map a say we then , ε k X f 7→ l ε + inf ֒ → ε ebdigfrom -embedding : k p n B 1 X K wdim } uhta hr xssan exists there that such 1 l ) balls p and, , dim ֒ − → ε ( n 1 ) / Y ( 2 p K ihthe with X ε hstp fmp,wihcan which maps, of type This . X ≤ ≤ . ∀ , . d ε ) ∈ eoe wspec denoted , ε ε ε neetdi h widths the in interested e a efud n n of one and found, be can srltdt h oinof notion the to related is R 7,w hl introduce: shall we [7], ε a h wdim the , > l < < n ∞ esthan less ( 0 X X , 2 2 erc( metric f ( k n oa to ) n h hl aaof data whole the e 3.I sthe is It [3]. see , : − − , X 1 M 1 / / n p p ( → ε -dimensional p -embedding ε . ε i.e. , , ( X Y ε R B wspec f )) i.e. ⊂ n 1 l san is p where , h sup the ( o the for Z n ) ∀ ≥ , y l 0 ∞ of ε X ∪ ∈ ε ) - l p(Γ), for Γ a countable group, with the natural action of Γ and the weak-∗ topology is zero when p < ∞ (see [14]). It is one of the possible ways of proving the non-existence of action preserving homeomorphisms between l∞(Γ) and l p(Γ); a simpler argument would be to notice that with the weak-∗ topology, Γ sends all points of l p(Γ) to 0 while l∞(Γ) has many periodic orbits. The behaviour is quite different when balls are looked upon with their natural met- ric. Theorem 1.4: Let p ∈ [1,∞), n > 1, then ∃hn ∈ Z satisfying hn = n/2 for n even, n+1 n−1 h3 = 2 and hn = 2 or 2 otherwise, such that p n {0,h(n),n}∪⊂ wspec(Bl (n),l p) ⊂{0} ∪ ( − 1,n] ∩ Z. 1 2 When p = 2 or when p = 1 and there is a Hadamard matrix of rank n + 1, then n − 1 p l (n) p also belongs to wspec(B1 ,l ). N n More precisely, let k,n ∈ with 2 − 1 < k < n. Then there exists bn;p ∈ [1,2] and ck,n;p ∈ [1,2) such that

p ε l (n) p if ≥ 2 then wdimε(B1 ,l )= 0 p ε l (n) p n if < 2 then wdimε(B1 ,l ) > − 1 p 2 ε l (n) p if ≥ ck,n;p then wdimε(B1 ,l ) ≤ k p ε l (n) p if < bk;p then wdimε(B1 ,l ) ≥ k and, for fixed n and p, the sequence ck,n;p is non-increasing. Furthermore, bk;p ≥ ′ 1/p′ 1 1/p 1/p 1 1/p ∞ 2 1 + k when 1 ≤ p ≤ 2, whereas bk;p ≥ 2 1 + k if 2 ≤ p < .

1 Additionally, in the Euclidean case (p = 2), we have that bn;2 = cn−1,n;2 = 2(1 + n ), 1/p 1/p′ q while in the 2-dimensional case b2;p ≥ max(2 ,2 ) for any p ∈ [1,∞]. Also, if p = 1 1, and there is a Hadamard matrix in dimension n + 1, then bn;1 = cn−1,n;1 = 1 + n . p ε l (n) 2 1/p
Finally, when n = 3, ∀ > 0,wdimεB1 6= 1 and c2,3;p ≤ 2( 3 ) , which means in particular that c2,3;p = b3;p when p ∈ [1,2]. Various techniques are involved to achieve this result; they will be exposed in sec- tion 3. While upper bounds on wdimεX are obtained by writing down explicit maps to a space of the proper dimension (these constructions use Hadamard matrices), lower bounds are found as consequences of the Borsuk-Ulam theorem, the filling radius of spheres, and lower bounds for the diameter of sets of n + 1 points not contained in an open hemisphere (obtained by methods very close to those of [9]). We are also able to give a complete description in dimension 3 for 1 ≤ p ≤ 2.

2 Properties of wdimε

Here are a few well established results; they can be found in [1], [2], [11], and [12]. Proposition 2.1: Let (X,d) and (X ′,d′) be two metric spaces. wdimε has the following properties:

a. If X admits a triangulation, wdimε(X,d) ≤ dimX.

2 b. The function ε 7→ wdimεX is non-increasing.

c. Let X be the connectedcomponentsof X, then wdimε X d 0 ε maxDiamX . i ( , )= ⇔ ≥ i i ′ ′ ′ d. If f : (X,d) → (X ,d ) is a continuousfunction such that d(x1,x2) ≤Cd ( f (x1), f (x2)) ∞ ′ ′ where C ∈]0, [, then wdimε(X,d) ≤ wdimε/C(X ,d ). e. Dilations behave as expected, i.e. let f : (X,d) → (X ′,d′) be an homeomorphism ′ such that d(x1,x2)= Cd ( f (x1), f (x2)); this equality passes throughto the wdim: ′ ′ wdimε(X,d)= wdimε/C(X ,d ).

f. If X is compact, then ∀ε > 0,wdimε(X,d) < ∞. Proof. They are brought forth by the following remarks:

a. IfdimX = ∞, the statement is trivial. For X a finite-dimensional space, it suffices to look at the identity map from X to its triangulation T(X), which is continuous and injective, thus an ε-embedding ∀ε. b. If ε ≤ ε′, an ε-embedding is also an ε′-embedding. ε ∋ c. If wdimεX = 0 then ∃φ : X ֒→ K where K is a totally discontinuous space. ∀k K,φ−1(k) is both open and closed, which implies that it contains at least one connected component, consequently DiamXi ≤ ε. On the other hand, if ε ≥ DiamXi the map that sends every Xi to a point is an ε-embedding. ′ ε φ ′ d. If wdimε/CX = n, there exists an C -embedding : X → K with dimK = n. Noticing that the map φ ◦ f is an ε-embedding from X to K allows us to sustain the claimed inequality. e. This statement is a simple application of the preceding for f and f −1.

f. To show that wdimε is finite, we will use the nerve of a covering; see [8, §V.9] for example. Given a covering of X by balls of radius less than ε/2, there exists, by compactness, a finite subcovering. Thus, sending X to the nerve of this finite covering is an ε-immersion in a finite dimensional polyhedron.

Another property worth noticing is that lim wdimε X d dimX for compact X; ε→0 ( , )= we refer the reader to [1, prop 4.5.1]. Reading [6, app.1] leads to believe that there is a strong relation between wdim and the quantities defined therein (Radk and Diamk); the existence of a relation between wdim and the filling radius becomes a natural idea, implicit in [7, 1.1B]. We shall make a small parenthesis to remind the reader of the definition of this concept, it is advised to look in [6, §1] for a detailed discussion. Let (X,d) be a compact metric space of dimension n,and let L∞(X) be the (Banach) space of real-valued bounded functions on X, with the norm k f kL∞ = sup | f (x)|. The x∈X metric on X yields an isometric embedding of X in L∞(X), known as the Kuratowski embedding: ∞ IX : X → L (X) ′ ′ x 7→ fx(x )= d(x,x ).

3 The triangle inequality ensures that this is an isometry: sup ′′ ′ ′′ ′ k fx − fx′ kL∞ = d(x,x ) − d(x ,x ) = d(x,x ). x′′ X ∈ ∞ Denote by Uε(X) the neighborhood of X ⊂ L (X) given by all points at distance less than ε from X, ∞ i.e. Uε X f L X inf f f ∞ ε ( )= ∈ ( ) x∈X k − xkL < .  Definition 2.2: The filling radius of a n-dimensional compact metric space X, written FilRadX, is defined as the smallest ε such that X bounds in Uε(X), i.e. IX (X) ⊂ Uε(X) induces a trivial homomorphism in simplicial homology Hn(X) → Hn(Uε(X)). Though FilRad can be defined for an arbitrary embedding, we will only be con- cerned with the Kuratowski embedding. Lemma 2.3: Let (X,d) bea n-dimensional compact metric space, k < n an integer, and Y ⊂ X a k-dimensional closed set representing a trivial (simplicial) homology class in Hk(X). Then ε < 2FilRadY ⇒ wdimε(X,d) > k.

If we remove the assumption that [Y ] ∈ Hk(X) be trivial, the inequality is no longer strict: wdimε(X,d) ≥ k.

Proof. Let us show that wdimε(X,d) ≤ k ⇒ ε ≥ 2FilRadY. Given an ε-embedding ε ≥ φ : X ֒→ K, then φ(Y ) ⊂ K bounds, since φ∗[Y]= 0 as [Y]= 0 in Hk(X). Since dimK k = dimY, the chain representing φ(Y ) is trivial. Compactness of X allows us to suppose that φ is onto a compact K. Otherwise, we restrict the target to φ(X). We will now ∞ produce a map Y → L (Y) whose image is contained in Uε/2(Y ), so that Y will bound ε ε in its 2 -neighborhood. This will mean that ≥ 2FilRadY. Let ∞ ∞ ∞ Q : K → L (X) ρY : L (X) → L (Y) ′′ ′′ ′ , and k 7→ gk(x )= ε/2 + inf d(x ,x ) x′∈φ−1(k) f 7→ f |Y . ρ φ δ First, notice that Y ◦ Q ◦ (Y) ⊂ Uδ+ε/2(Y ),∀ > 0 : ε ρ φ sup inf ′′ ′ ′′ ε k Y ◦ Q ◦ (y) − IY (y)kL∞ = + d(y ,y ) − d(y ,y) = /2, y′′∈Y 2 y′∈φ−1(φ(y)) 

since φ is an ε-embedding. Second, (ρY ◦ Q ◦ φ)∗[Y ]= 0 and (ρY ◦ Q ◦ φ) ∼ IY in ∞ ε Uδ+ε/2(Y ), as L (Y ) is a vector space. Consequently, [IY (Y)] = 0 and ≥ 2FilRadY, by letting δ → 0. If [Y ] 6= 0 ⊂ Hk(X), the proof still follows by taking K of dimension k − 1: the homology class φ∗[Y ] is then inevitably trivial, since K has no rank k homology. Thus, calculating FilRad is a good starting point. The following lemma consists of a lower bound for FilRad: Lemma 2.4: Let X be a closed convex set in a n-dimensional normed vector space. Suppose it contains a point x0 such that the convex hull of n + 1 points on ∂X whose diameter is < a excludes x0. Then FilRad∂X ≥ a/2, and, using lemma 2.3, ε < a ⇒ wdimεX = n.

4 Proof. Suppose that Y = ∂X has a filling radius less than a/2. Then, ∃ε > 0 and ∃P a polyhedron such that Y bounds in P, P ⊂ U a ε(Y ) and that the simplices of P have a 2 − diameter less than ε. To any vertex p ∈ P it is possible to associate f (p) ∈ IY (Y) so that a ε kp, f (p)kL∞(Y) < 2 − . Let p0,..., pn be a n-simplex of P, a Diam{ f (p ),..., f (p )} < 2( − ε)+ ε < a − ε < a. 0 n 2

Since IY is an isometry, f (pi) can be seen as points of Y without changing the diameter of the set they form. The convex hull of these f (pi) in B will not contain x0: their distance to f (p0) is < a which excludes x0. Let π be the projection away from x0, that is associate to x ∈ X, the point π(x) ∈ ∂X on the half-line joining x0 to x. Using π, the n-simplex generated by the f (pi) yields a simplex in Y . Thus we extended f to a retraction r from P to Y . Let c be a n-chain of P which bounds Y, i.e. [Y]= δc. A contradiction becomes apparent: [Y ]= r∗[Y ]= r∗δc = δr∗c. Indeed, if that was to be true, Y , which is n − 1 dimensional would be bounding an n-dimensional chain in Y. Hence FilRadY > a/2. This yields, for example: Lemma 2.5: (cf. [7, 1.1B]) Let B be the unit ball of a n-dimensional Banach space, then ∀ε < 1,wdimεB = n. Proof. Any set of n+1 points on Y = ∂B whose diameter is less than 1 does not contain the origin in its convex hull. So according to lemma 2.4, FilRadY > 1/2, and since Y is a closed set of dimension n−1 whose homology class is trivial in B, we conclude by applying lemma 2.3. Let us emphasise this important fact on l∞ balls in finite dimensional space. ∞ l (n) n Rn Lemma 2.6: Let B1 = [−1,1] be the unit cube of with the product (supremum) metric, then ∞ l (n) 0 if ε ≥ 2 wdimεB = . 1  n if ε < 2 This lemma will be used in the proof of proposition 1.3. Its proof, which uses the Brouwer fixed point theorem and the Lebesgue lemma, can be found in [12, lem 3.2], [2, prop 2.7] or [1, prop 4.5.4].

Proof of proposition 1.3: We first show the lower boundon wdimε. Ina k-dimensional ∞ p ∞ −1/p p l (k) l (k) space, the l ball of radius k is included in the l ball: B ⊂ B , as kxk p ≤ k−1/p 1 l (k) p p p 1/p l (k) l (n) l (n) k kxkl∞(k). Since B1 ⊂ B1 , by 2.1.d, we are assured that, if B1 is considered p ∞ ε −1/p l (n) ∞ with the l metric, < 2k implies that wdimε(B1 ,l ) ≥ k. To get the upper bound, we give explicit ε-embeddings to finite dimensional poly- hedra. This will be done by projecting onto the union of (n − j)-dimensional coordi- nates hyperplanes (whose points have at least j coordinates equal to 0). Project a point p l (n) π th x ∈ B1 by the map j as follows: let m be its j smallest coordinate (in absolute value), set it and all the smaller coordinates to 0, other coordinates are substracted m if they are positive or added m if they are negative.

5 ε n ε ε Denote by~ an element of {−1,1} and~ \A the same vector in which ∀i ∈ A, i is replaced by 0. The largest fibre of the map π j is

p −1 l (n) π (0)= ∪ {λ0~ε + ∑ λl~ε i i |λi ∈ R≥0} ∩ B . j ε \{ 1,..., l} 1 ~ ,i1,...,i j−1 1≤l≤ j−1

−1/p −1/p Its diameter is achieved by s0 = (n− j+1) ,...,(n− j+1) ,0,...,0 and −s0; π−1 −1/p π ε −1/p thus Diam j (0)= 2(n− j+1) . j allows us to assert that > 2(n− j+
1) ⇒ l p(n) ∞ wdimε(B1 ,l ) ≤ n− j, by realising a continuousmap in a (n− j)-dimensional poly- hedron whose fibres are of diameter less than 2(n − j + 1)−1/p.

p The above proof for an upper bound also gives that wdimε(Bl (n),lq) ≤ k if ε ≥ 2(k + 1)1/q−1/p, but the inclusion of a lq ball of proper radius in the l p ball gives a lower bound that does not meet these numbers. Also note that lemma 2.3 is efficient to evaluate width of tori, as the filling radius of a product is the minimum of the filling radius of each factor.

l p(n) 3 Evaluation of wdimεB1

We now focus on the computation of wdimεX for unit ball in finite dimensional l p. Except for a few cases, the complete description is hard to give. We start with a simple example. 1 Example 3.1: Let Bl (2) be the unit ball of R2 for the l1 metric, then ε l1(2) 0 if ≥ 2, wdimεB =  2 if ε < 2.

1 1 If Bl (2) is endowed with the l p metric, then ε < 21/p ⇒ wdimεBl (2) = 2. Proof. Given any three points whose convex hull contains the origin, two of them have to be on opposite sides, which means their distance is 21/p in the l p metric. Hence a radial projection is possible for simplices whose vertices form sets of diameter less 1 than 21/p. Invoking lemma 2.4, FilRad∂Bl (2) ≥ 2−1+1/p. Lemma 2.3 concludes. This is specific to dimension 2 and is coherent with lemma 2.6, since, in dimension 2, l∞ and l1 are isometric. An interesting lower bound can be obtained thanks to the Borsuk-Ulam theorem; as a reminder, this theorem states that a map from the n-dimensional sphere to Rn has a fibre containing two opposite points. ∂ l p(n+1) Proposition 3.2: Let S = B1 be the unit sphere of a (n+1)-dimensional Banach space, then ε < 2 ⇒ wdimεS > (n − 1)/2. l p(n+1) ε l p(n+1) In particular, the same statement holds for B1 : < 2 ⇒ wdimεB1 > (n − 1)/2.

6 n−1 Proof. We will show that a map from S to a k-dimensional polyhedron, for k ≤ 2 , sends two antipodal points to the same value. Since radial projection is a homeomor- 2 n ∂ l (n+1) phism between S and the Euclidean sphere S = B1 that sends antipodal points to antipodal points, it will be sufficient to show this for Sn. Let f : Sn → K be an ε- embedding, where K is a polyhedron, dimK = k ≤ (n − 1)/2 and ε < 2. Since any polyhedron of dimension k can be embedded in R2k+1, f extends to a map from Sn to Rn that does not associate the same value to opposite points, because ε < 2. This contradicts Borsuk-Ulam theorem. The statement on the ball is a consequence of the inclusion of the sphere.

l p(n) n Hence, wdimεB1 always jumps from 0 to at least ⌊ 2 ⌋ if they are equipped with their proper metric.

A first upper bound. Though this first step is very encouraging, a precise evaluation of wdim can be convoluted, even for simple spaces. It seems that describing an explicit continuous map with small fibers remains the best way to get upper bounds. Denote by n = {0,...,n}. Lemma 3.3: Let B be a unit ball in a normed n-dimensional real vector space. Let {pi}0≤i≤n be points on the sphere S = ∂B that are not contained in a closed hemisphere. n λ R n λ Suppose that ∀A ⊂ with |A|≤ n−2, and ∀ j ∈ >0, where j ∈ , if k∑i∈A i pik≤ 1, λ λ λ k ∈/ A and k∑i∈A i pi − k pkk≤ 1, then k k pkk≤ 1. A set pi satisfying this assumption gives ε ≥ Diam{p } := max p − p ⇒ wdimεB ≤ n − 1. i i6= j i j

Proof. This will be done by projecting the ball on the cone with vertex at the origin over the n − 2 skeleton of the simplex spanned by the points pi. Note that n + 1 points satisfying the assumption of this lemma cannot all lie in the same open hemisphere, however we need the stronger hyptothesis that they do not belong to a closed hemi- sphere. Now let ∆n be the n-simplex given by the convex hull of p0,..., pn. We will project the ball on the various convex hulls of 0 and n − 1 of the pi. Call E the radial projection of elements of the ball (save the origin) to the sphere, and let, for A ⊂ n, PA = {p0,..., pn}\{pi|i ∈ A}. In particular, P∅ is the set of all the pi. Furthermore, denote by CX the convex hull of X. Given these notations, ECP{i} is the radial pro- jection of the (n − 1)-simplex CP{i} (CP{i} does not contain 0 else the points would lie in a closed hemisphere), and ECP{i, j} are parts of the boundary of this projection. n ∆′ C EC Finally, consider, again for A ⊂ , A = [ PA ∪ 0]. ′ ′ Let si : ∆ → ∪ ∆ be the projection along pi. More precisely, we claim that {i} j6=i {i, j} ∆′ Λ λ λ R si(p) is the unique point of {i, j} that also belongs to pi (p)= {p + pi| ∈ ≥0}. Existence is a consequence of the fact that the points are not contained in an closed R ∑ ∆′ ∆′ hemisphere, i.e. ∃µi ∈ >0 such that k∈n µk pk = 0. Indeed, p ∈ {i}, if p ∈ {i, j} ∆′ for some j, then there is nothing to show. Suppose that ∀ j 6= i, p ∈/ {i, j}. Then p = ∑ λ p , where λ > 0. Write p = − 1 ∑ µ p . It follows that for some λ, p + k6=i k k k i µi k6=i k k λ ∑ λ′ λ′ λ pi can be written as k∈n\{i, j} k pk with 0 ≤ k ≤ k. Uniqueness comes from a ∆′ transversality observation. {i, j} is contained in the plane generated by the set P{i, j}

7 Λ and 0 which is of codimension 1. If the line pi (p) was to lie in that plane then the set P{ j} would lie in the same plane, and P∅ would be contained in a closed hemisphere. Λ ∆′ ∆′ Thus pi (p) is transversal to {i, j}. The figure below illustrates this projection in {0} for n = 3. ∆ {0,3} p p 2 ∆ {0,1}

p 1 p 3

0 ε Our (candidate to be an) -embedding s is defined by s|∆′ = si. Since on ECP{i} ∩ {i} ∆′ ECP{ j} ⊂ ECP{i, j}, we see that s|∆′ = Id and that ∪ = B, this map is well- {i, j} i∈n {i} defined. It remains to check that the diameter of the fibres is bounded by ε. We claim −1 that the biggest fibre is s (0)= ∪iC{−pi,0}, whose diameter is that of the set of ∆′ vertices of the simplex, Diam{pi}. To see this, note that for x ∈ {i, j}, the diameter of s−1(x) attained on its extremal points (by convexity of the norm), that is x and points of λ ∆′ ∆′ the form x − k pk (for k ∈ A, where A ⊃{i, j} and x ∈ A ⊂ {i, j}) whose norm is one. However, since x = ∑λi pi for i ∈/ A and λi > 0, kx − λk pkk = 1 implies kλk pkk≤ 1, so a simple translation of s−1(x) is actually included in s−1(0). This allows us to have a first look at the Euclidean case. 2 l (n) Rn Theorem 3.4: Let B1 be the unit ball of , endowed with the Euclidean metric, 1 and let bn;2 := 2(1 + n ). Then, for 0 < k < n, q l2(n) ε wdimεB1 = 0 if 2 ≤ , l2(n) ε k ≤ wdimεB1 < n if bk+1;2 ≤ < bk;2, l2(n) ε wdimεB1 = n if < bn;2.

ε l2(n) Proof. First, when ≥ DiamB1 = 2 this result is a simple consequence of proposi- tion 2.1.c; when n = 1 it is sufficient, so suppose from now on that n ≥ 2. Applying ∂ l2(n) l2(n) l2(n) ε ∂ l2(n) lemma 2.3 to B1 ⊂ B1 yields that wdimεB1 = n if < 2FilRad B1 , but l2(n) FilRadB1 ≥ bn;2 by Jung’s theorem (see [4, §2.10.41]), as any set whose diameter l2(n) is less than < bn;2 is contained in an open hemisphere ([10] shows that FilRadB1 = l2(n) bn;2). On the other hand, balls of dimension k < n are all included in B1 , which l2(k) l2(n) l2(n) means that wdimεB1 ≤ wdimεB1 , thanksto 2.1.d. Hencewe havethatwdimεB1 ≥ k whenever bk+1;2 ≤ ε < bk;2. This proves the lower bounds.

8 The vertices of the standard simplex satisfy the assumption of lemma 3.3: thanks to the invariance of the norm under rotation we can assume p0 = (1,0,...,0). The other pi will all have a negative first coordinate, and so will any positive linear combination. Substracting λp0 will be norm increasing. As the diameter of this set is bn;2, lemma 3.3 gives the desired upper bound. Let us now give an additional upper bound for the 3-dimensional case: p ∞ ε 2 1/p l (3) Proposition 3.5: If 1 ≤ p < , then ≥ 2( 3 ) ⇒ wdimεB1 ≤ 2. Proof. In R3 there is a particularly good set of points to define our projections. These - 1 - 1 - 1 - 1 are p0 = 3 p (1,1,1), p1 = 3 p (1,-1,-1), p2 = 3 p (-1,1,-1) and p3 = 3 p (-1,-1,1). Let λ λ λ λ λ R x = 1 p1, where ∈ [0,1], and suppose k 1 p1 − 2 p2kl p ≤ 1 for 2 ∈ ≥0. We have to λ λ λ λ 2 λ λ p 1 λ check that 2 ≤ 1. Suppose 2 > 1, then 1 ≥k 1 p1 − 2 p2kl p = 3 ( 1 + 2) + 3 ( 2 − λ p λp 2 p 1 p λ λ 1) = 2 [ 3 (1+t) + 3 (1−t) ], where t = 1/ 2. The functionof t has minimal value 1, which gives λ2 ≤ 1 as desired. Suppose now that x = λ1 p1 + λ2 p2 is of norm less than 1, where without loss of λ λ λ λ λ generality we assume 2 ≥ 1, and k 1 p1 + 2 p2 − 3 p3kl p ≤ 1. kxkl p ≤ 1 implies that 1 λ λ p 2 λ λ p λ λ p 1 λ λ p 1 λ λ p 1 ≥ 3 ( 1 + 2) + 3 ( 2 − 1) so ( 2 − 1) ≤ 1 − 3 ( 2 + 1) + 3 ( 2 − 1) ≤ 1. If λ3 > 1, then λ λ λ 1 ≥k 1 p1 + 2 p2 − 3 p3kl p 1 λ λ λ p 1 λ λ λ p 1 λ λ λ p = 3 ( 3 + 2 + 1) + 3 ( 3 − ( 2 − 1)) + 3 ( 3 + ( 2 − 1)) . However,

λp 1 λ λ λ p 2 λp 3 ≤ 3 ( 3 + 2 + 1) + 3 3 1 λ λ λ p 1 λ λ λ p 1 λ λ λ p ≤ 3 ( 3 + 2 + 1) + 3 ( 3 − ( 2 − 1)) + 3 ( 3 + ( 2 − 1)) ≤ 1.

Using that f (t) = (1 + t)p + (1 − t)p has minimum 2 for t ∈ [0,1]. These arguments can be repeated for any indices to show that the points pi, where i = 0,1,2 or 3, satisfy the assumption of lemma 3.3. The conclusion follows by showing that Diam(pi)= 2 1/p 2( 3 ) For certain dimensions, a set of points that allows to build projections with small fibers can be found. Their descriptions require the concept of Hadamard matrices of rank N; these are N × N matrices, that will be denoted HN , whose entries are ±1 and t such that HN · HN = NId. It has been shown that they can only exist when N = 2 or 4|N, and it is conjectured that this is precisely when they exist. Up to a permutation and a sign, it is possible to write a matrix HN so that its first column and its first row consist only of 1s. It is quite easy to see that two rows or columns of such a matrix have exactly N/2 identical elements. Definition 3.6: Let HN be a Hadamard matrix of rank N, and let, for 0 ≤ i ≤ N, hi th be the i row of the matrix without its first entry (which is a 1). Then the hi form a Hadamard set in dimension N − 1.

9 These N elements, normalised so that khikl p(N−1) = 1. When so normalised, their p 1−1/p 1 p ∑ diameter (for the l metric) is 2 (1 + N−1 ) . Since hi = 0, by orthogonality of the columns with the column of 1 that was removed, we see that they are not contained in an open hemisphere. The set of points in the preceding proposition was given by a Hadamard matrix of rank 4, and when p = 2 the convex hull of these points is just the standard simplex. Proposition 3.7: Suppose there exists a Hadamard matrix of rank n + 1, then

1 l1(n) ε ≥ 1 + ⇒ wdimεB ≤ n − 1. n 1

N Proof. Let the hi be as above, and N = n + 1. Note that for i 6= j, hi and h j have 2 N λ λ opposed coordinates, and 2 − 1 identical ones. Thus ihi − jh j has always a big- 1 ger l norm than any of its two summands. Indeed, the coefficients c j of the vector ∑ λihi where the contribution of hk reduces c j are in lesser number than those that i∈A 1 get increased. Since the l norm is linear, the magnitude of the c j getting smaller is not relevant, only their number. 1 We conclude by applying lemma 3.3, as Diaml1 (hi)= 1 + N−1 . Note that in dimension higher than 3 and for p > 2, Hadamard sets no longer satisfy the assumption of lemma 3.3.

l p(n) Further upper bounds for wdimεB1 . The projection argument still works for non-Euclidean spheres. It can also be repeated, though unefficiently, to construct maps to lower dimensional polyhedra. ∞ l p(n) Proposition 3.8: For 1 < p < , consider the sphere B1 with its natural metric. n−1 Then, for 2 < k < n, ∃ck,n;p ∈ [1,2) such that ck,n;p ≥ ck+1,n;p, and

l p(n) ε wdimεB1 ≤ k if ≥ ck,n;p.

Furthermore cn−1,n;2 = bn;2 Proof. This proposition is also obtained by constructing explicitly maps that reduce n+1 dimension (up to n− j for j < 2 ) and whose fibres are small. Unfortunately, nothing indicates this is optimal, and the size of the preimages is hard to determine. We will l p(n) abbreviate B := B1 . We proceed by induction, and keep the notations introduced in the proof of lemma 3.3. The pi that are used here are the vertices of the simplex; they need to be renor- malised to be of l p-norm 1, but note that multiplying them by a constant has actually ∆′ no effect in this argument. Also note that the sets A are not the same for different p, since they are constructed by radial projection to different spheres. The keys to this construction are the maps s : ∆′ → ∪ ∆′ given by j;{i1,...,i j} {i1,...,i j} {i1,...,i j,m} m∈{/ i1,...,i j} j ∑ σ projection along the vectors pil . Call 1 the function s from lemma 3.3, then, for l=1

10 ′ j > 1, σ j : B → ∪ ∆ is obtained by composing, on appropriate do- {i1,...,i j+1} {i1,...,i j+1}⊂n σ mains, s j;{i1,...,i j} with j−1. Since s j;i1,...,i j are equal to the identity when their domain intersect, and their union covers the image of σ j−1, the map is again well-defined. It remains only to calculate the diameter of the fibres. At 0 the fibre is σ−1 λ λ λ λ λ λ R j (0)= ∪ {−( 1 +...+ j)pi1 −( 1 +...+ j−1)pi2 −...− 1 pi j | i ∈ ≥0}. {i1,...,i j}⊂n

∆′ Whereas for a given x ∈ A in the image (that is A contains at least j elements), x can also be written down as a combination ∑λi pi, for i ∈/ A and λi ∈ R>0. We have σ−1 λ λ λ λ λ λ R j (x)= ∪ {x−( 1 +...+ j)pi1 −( 1 +...+ j−1)pi2 −...− 1 pi j | i ∈ ≥0}. {i1,...,i j}⊂A

2 sup σ−1 ε l (n) If we set ck,n = Diam n−k(x), then when ≥ ck,n, wdimεB1 ≤ k. It is possible x∈σ j(B) to determine two simple facts about these numbers. First, they are non-increasing ck,n ≥ ck+1,n, which is obvious as the construction is done by induction, the size of the fiber of maps to lower dimension is bigger than for maps to higher dimension. Second, they are meaningful: ck,n < 2. Indeed, when p 6= 1,∞, ck,n = 2 only if σ−1 n−k(x) contains opposite points, which is a linear condition. When x 6= 0, by convexity of the distance, the points on which the diameter can be attained are at the boundary of σ−1 j (x). Say Y is the set of those point except x. The distance from Y to x is at most one, while the diameter of Y is bounded. Indeed, there is a cap of diameter less than 2 that contains all the pi but one. The biggest diameter of such caps is also less than 2 and bounds DiamY . Any pointof the fibre at 0 is a linear combinationof the vertices pi, and thereis only ∑ n+1 n−1 one linear relation between these, namely pi = 0. As long as j < 2 (i.e. k > 2 ) σ−1 there are not enough pi in any two sets that form j (0) to combine into the required relations, but as soon as j exceeds this bound, opposite points are easily found. l p(n) ∞ For B1 , where 1 < p < , we used the regular simplex to describe our projec- tions, though nothing indicates that this choice is the most appropriate. In fact, many sets of n+1 points allow to build projections to a polyhedron, but it is hard to tell which are more effective: on one hand we need this set to have a small diameter (so that the fibre at 0 is small), while on the other, we need it to be somehow well spread (so as to avoid fibres at x to be too large, as in the assumption of lemma 3.3). Furthermore, there is in general no reason for cn−1,n;p to coincide with a lower bound, or even to be p l (n) p different from other ck;p, thus we cannot always insure that n − 1 ∈ wspec(B1 ,l ).

The lowest non zero element of wspec. Before we return to the general l p case, no- tice that together proposition 3.2 and theorem 3.4 give a good picture of the function l2(n) ε ε wdimεB1 . It equals n for < bn;2 = cn−1,n;2, then n − 1 for bn;2 ≤ < bn−1;2. After- wards, I could not show a strict inequality for the ck,n;2, but even if they are all equal, l2(n) n n Z ε wdimεB1 takes at least one value in ( 2 − 1, 2 + 1) ∩ . Then ,when ≥ 2, it drops to 0.

11 For odd dimensional balls, there is a gap between the value given by proposition 3.2 and the lowest dimension obtained by the projections introduced above. Say B is of dimension 2l + 1 and ε less than but sufficiently close to 2, then on one hand we know that wdimεB ≥ l, while on the other wdimεB ≤ l + 1. It is thus worthy to ask whether one of these two methods can be improved, perhaps by using extra homological in- formation on the simplices in the proof of proposition 3.2 (e.g. if its highest degree cohomology is trivial then a k-dimensional polyhedron is embeddable in R2k, see [5]). Remark 3.9: Such an improvementis actually available when n = 3: if the 2-dimensional sphere maps to a 1-dimensional polyhedron (i.e. a graph), the map lifts to the universal cover, a tree K. Hence K is embeddable in R2, and, for 1 < p < ∞.

ε l p(3) < 2 ⇒ wdimεB1 ≥ 2 for otherwise it would contradict Borsuk-Ulam theorem. Note that estimates obtainedin [6, app 1.E5]for Diam1, can also yield lower bounds for the diameter of fibres for maps to graphs (i.e. 1-dimensional polyhedra). Applied to spheres, it becomes a special case of proposition 3.2 and of the above remark.

l p(n) Lower bounds for wdimεB1 . The remainder of this section is devoted to the im- provement of lower bounds, using an evaluation of the filling radius as a product of lemma 2.4, and a short discussion of their sharpness. We shall try to find a lower bound on the diameter of n + 1 points on the l p unit sphere that are not in an open hemisphere; recall that points fi are not in an open hemisphere if ∃λi such that ∑λi fi = 0. A direct use of Jung’s constant (defined as the supremum over all convex M of the radius of the smallest ball that contains M divided by M’s diameter) that is cleverly estimated for l p spaces in [9] does not yield the result like it did in the Euclideancase. This is due to the fact that there are sets of n+1 points on the sphere that are not contained in an open hemisphere, but are contained in a ball (not centered at the origin) of radius less than 1. The set of points given by

2 2 2 2 (3.10) (1,...,1), − ,...,− ,1 ,..., and 1,− ,...,−  n − 1 n − 1   n − 1 n − 1 is such an example for l∞, and deforming it a little can make it work for the l p case, p finite but close to ∞. However, a very minor adaptation of the methods given in [9] is sufficient. First, we introduce norms for the spaces of sequences (and matrices) taking val- n ues in a Banach space E. Let αi ∈ R≥0 be such that ∑ αi = 1 and denote by α this i=0 sequence of n + 1 real numbers. Let Ep,α be the space of sequences made of n + 1 ele- ments of E and consider the l p norm weighted by α: x ∑ α x p 1/p where k kEp,α = i i k ikE x = (x0,...,xn). On the other hand, Ep,α2 shall represent the space of matrices
whose ∑ α α p 1/p entries are in E, with the norm (xi, j) E = i, j i j xi, j E . Now define, for p,α2 ′
E,E Banach spaces based on the same vector space and for 1 ≤ s,t ≤ ∞, the linear operator T : E α E′ by x x x . s, → t,α2 ( i) 7→ ( i − j)

12 Theorem 3.11: Consider a vector space on which two norms are defined, and denote ∗ by E1, E2 the Banach space they form. Let fi ∈ E , 0 ≤ i ≤ n, be such that k fik ∗ = 1 1 E1 but that they are not included in an open hemisphere, i.e. there exists λi ∈ R≥0 such λ λ sup that ∑ i fi = 0 and ∑ i = 1. Let DiamE∗ ( f )= fi − f j ∗ be the diameter of 2 0≤i, j≤n E2 this set with respect to the other norm. Then, for αi = λ i,

1/t′ 1 −1 Diam ∗ ( f ) ≥ 2 sup 1 + sup kT k . E2 (E1)s,α→(E2) 2 1≤s,t≤∞  n E1 t,α

Proof. As the fi are not in an open hemisphere, real numbers λi ∈ R≥0 such that ∑λi = 1 and ∑λi fi = 0 exist. Furthermore, since k fik ∗ = 1, there also exist xi ∈ E1 such that E1 f (x )= 1 and kx k = 1. The remark on which the estimation relies is, as in [9], i i i E1 n 2 = ∑ λiλ j( fi − f j)(xi − x j). i, j=0

Choosing αi = λi, this equality can be rewritten in the form 2 = (T f )(Tx), where ∗ ∗ Tx ∈ (E2)t,α2 and T f ∈ ((E2)t,α2 ) = (E )t′,α2 , and thus 2 ≤kT f k E∗ kTxk E . 2 ( 2 )t′,α2 ( 2)t,α2 Notice that n 1 n ∑αiα j = ∑αi(1 − αi) ≤ 1 − = , i6= j i=0 n + 1 n + 1 α α −1/2 because k ikl1(n+1) = 1 ⇒k ikl2(n+1) ≥ (n + 1) . We can isolate the required di- ameter: n t′ 1/t′ kT f k ∗ = ∑ α α f − f ∗ (E ) ′ 2 i j i j E 2 t ,α i=0 2
1/t′ ≤ Diam ∗ ( f ) ∑ α α E2 i j i6= j ′
n 1/t ≤ Diam ∗ ( f ) . E2 n+1
On the other hand, kx k = 1, consequently kxk = 1, so we bound i E1 (E1)s,α

kTxk E ≤kT k E α E . ( 2)t,α2 ( 1)s, →( 2)t,α2 The conclusion is found by substitution of the estimates for the norms of T f and Tx.

We only quote the next result, as there is no alteration needed in that part of the argument of Pichugov and Ivanov. Theorem 3.12: (cf. [9, thm 2])

′ 1/p n 1/p−1/p if 1 ≤ p ≤ 2, kT k l p n ∞ α l p n ≤ 2 , ( ( )) , →( ( ))p,α2 n+1 ∞ 1/p′
if 2 ≤ p ≤ , kT k l p n ∞ α l p n ≤ 2 . ( ( )) , →( ( ))p,α2

p A simple substitution in theorem 3.11, with E1 = E2 = l (n), s = ∞ and t = p, yields the desired inequalities.

13 p Corollary 3.13: Let fi, 0 ≤ i ≤ n, be points on the unit sphere of l (n) that are not included in an open hemisphere, then

1/p′ 1 1/p if 1 ≤ p ≤ 2, Diaml p(n)( f ) ≥ 2 1 + n , (∗) ′ ∞ 1/p 1
1/p if 2 ≤ p ≤ , Diaml p(n)( f ) ≥ 2 1 + n . (∗∗)
Remark 3.14: Before we turn to the consequences of this result on wdimε, note that there are examples for which the first inequality is attained. These are the Hadamard sets defined in 3.6. When normalised to 1, they are not included in an open hemisphere and of the proper diameter. Hence, when a Hadamard matrix of rank n + 1 exists, then (∗) is optimal. Nothing so conclusive can be said for other dimensions, see the argumentin example 3.1. I ignore if there are cases for which (∗∗) is optimal, though it p l (n) n+1 1/p ∞ is very easy to construct a family Fn ∈ (B1 ) such that DiamFn → 2 as n → . ∞ −1 In particular for p = , the points given in (3.10) but by substituting n−1 instead of the −2 entries with value n−1 , is a set that is not contained in an open hemisphere and whose n diameter is n−1 , which is close to the bound given. Somehow, this case, is also the one where the use of lemma 2.4 results in a bound that is quite far from the right value of wdim, cf. lemma 2.6. This might not be so surprising as sets with small diameter on l p balls seem, when p > 2, to differ from sets satisfying the assumption of lemma 3.3. Still, by lemma 2.4 we obtain the following lower bounds on wdim: 1/p′ 1 1/p Corollary 3.15: Let bk;p be definedby bk;p = 2 1 + k when 1 ≤ p ≤ 2, whereas ′ 1/p 1 1/p ∞
bk;p = 2 1 + k if 2 ≤ p < . Then, for 0 < k ≤ n,
ε l p(n) < bk;p ⇒ wdimεB1 ≥ k. ∂ l p(n) Proof. Let Y = B1 . Since the convex hull of a set of n + 1 points on the sphere Y will not contain the origin if the diameter of the set is larger than bn;p, lemma 2.4 ensures that FilRadY ≥ bn;p/2. We then use lemma 2.3 for Y to conclude. These inequations might not be optimal, proposition 3.2 for example is always stronger when k < ⌊ n ⌋. 2 ∞ l (n) l p(n) l p(n) In dimension n, B ⊂ B yields that ε < 2n−1/p ⇒ wdimεB = n which n−1/p 1 1 improves corollary 3.15 as long as

2n2 ln( n+1 ) p ≥ 2n . ln( n+1 )

However, when p = 1, and Hn+1 is a Hadamard matrix, these estimates are as sharp as we can hope since the lower bound meets the upper bounds. Corollary 3.16: Suppose there is a Hadamard matrix of rank n + 1. Then, for 0 ≤ k < n, l1(n) ε wdimεB1 = 0 if 2 ≤ , n 1 l1(n) 1 1 − ε ε max( 2 ,k) ≤ wdim B1 < n if (1 + k+1 ) ≤ < (1 + k ), l1(n) ε 1 wdimεB1 = n if < (1 + n ).

14 Furthermore, in dimension 3, lower bounds of corollary 3.15 meet upper bounds of proposition 3.5 when 1 ≤ p ≤ 2. In particular, thanks to remark 3.9, this gives a complete description of the 3-dimensional case for such p. Corollary 3.17: Let p ∈ [1,2], then

0 if 2 ≤ ε, p l (3) 2 1/p ε wdimεB1 =  2 if 2( 3 ) ≤ < 2,  ε 2 1/p 3 if < 2( 3 ) .  ε l p(3) When p > 2, all that can be said is that the value of for which wdimεB1 drops 2 1−1/p 2 1/p from 3 to 2 is in the interval [2( 3 ) ,2( 3 ) ]. This last corollary is special to the 3-dimensional case, which happens to be a dimension where there exist a Hadamard set, and where the Borsuk-Ulam argument n−1 can be improved to rule out maps to 2 -dimensional polyhedra. For example, in the 2-dimensional case, a precise description is not so easy. Indeed, thanks to example 3.1 l1(2) l p(2) l p(2) and using the inclusion of B ⊂ B , we knowthat wdimεB = 2 when ε ≥ 21/p. 1 1 ∞ 1 l (2) l p(2) ′ l p(2) On the other hand, the inclusion of B ⊂ B gives ε ≥ 21/p ⇒ wdimεB = 2. 2−1/p 1 1 Putting these together yields:

p ε 1/p 1/p′ l (2) ≥ max(2 ,2 ) ⇒ wdimεB1 = 2. These simple estimates in dimension 2 are better than corollary 3.15 as long as p ≤ ln3 8 4 3 − ln2 or p ≥ ln( 3 )/ln( 3 ). I doubt that any of these estimations actually gives the ε l p(2) value of where wdimεB1 drops from 2 to 1. All the results of this section can be summarised to give theorem 1.4. Here are two depictions of the situation. Gray areas correspond to possible values, full lines to known values and dotted line to bounds. wdm ε wdm ε

n



n































n



n













2 2

b c 2 ε b c 2 ε {n;p} {n;p} The left-hand plot is for euclidean case, or the case where p = 1 and there is a Hadamard set, when the dimension is odd and different from 3: this is when a map to a n − 1-dimensional polyhedron with small fibers can be constructed, but the bounds from the Borsuk-Ulam argument and projections to lower dimensional polyhedron do not meet. The right-hand one represents the situation in cases where the dimension is

15 even and there is no known projection with small fibers. c⌈n/2⌉,n;p is abbreviated by c. The case of dimension 3 is described in corollary 3.17. n−1 It is not expected that 2 be in wspec when n is odd, nor is it expected that the l p(n) lower bounds bk;p be sharp for B1 when k < n.

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